**Simplified problem**: Given a $d$-times nested convolution of an input function $g(x):\mathbb{R}\mapsto \mathbb{R}$ with the same band-limited smooth function $f(x):\mathbb{R}\mapsto \mathbb{R}$. I am interested in the value of this $d$-dimensional integral $I$

$I=<f*<f*<...<f*g>..>$,

which I compute using Monte Carlo (this is a necessary evil for the non-simplified problem).

**Question:** Given that we can estimate spectrum of both $f$ and $g$, how to construct the best importance function for the MC integration?
In other words, how to distribute sampling density to different integration dimensions?

Intuitively, I would probably sample the outermost convolution most often, since its integrand has the largest spectral bandwidth (each convolution is a smoothing of the initial function).